Commutators as products of squares

Abstract:

It is shown that if $G$ is the free group of rank 2 freely generated by $x$ and $y$, then $xy{x^{ - 1}}{y^{ - 1}}$ is never the product of two squares in $G$, although it is always the product of three squares in $G$. It is also shown that if $G$ is the free group of rank $n$ freely generated by ${x_1},{x_2}, \cdots ,{x_n}$, then $x_1^2x_2^2 \cdots x_n^2$ is never the product of fewer than $n$ squares in $G$.